Enomoto and Ota's conjecture holds for large graphs

TL;DR

This paper proves Enomoto and Ota's conjecture for large graphs, showing that under certain degree conditions, specified path partitions exist with prescribed endpoints and lengths.

Contribution

The paper confirms the conjecture for large graphs using the Regularity Lemma and extremal techniques, advancing understanding of graph path partitions.

Findings

Conjecture holds for sufficiently large graphs.

Path partitions with prescribed endpoints and lengths exist under degree conditions.

Uses Regularity Lemma and extremal methods in proof.

Abstract

In 2000, Enomoto and Ota conjectured that if a graph $G$ satisfies $\sigma_{2}(G) \geq n + k - 1$, then for any set of $k$ vertices $v_{1}, \dots, v_{k}$ and for any positive integers $n_{1}, \dots, n_{k}$ with $\sum n_{i} = |G|$, there exists a partition of $V(G)$ into $k$ paths $P_{1}, \dots, P_{k}$ such that $v_{i}$ is an end of $P_{i}$ and $|P_{i}| = n_{i}$ for all $i$. We prove this conjecture when $|G|$ is large. Our proof uses the Regularity Lemma along with several extremal lemmas, concluding with an absorbing argument to retrieve misbehaving vertices.